Calculating Measures Of Supplementary Adjacent Angles AOB & BOC
Hey guys! Today, we're diving into a fun geometry problem involving supplementary adjacent angles. It might sound a bit intimidating at first, but trust me, we'll break it down step by step so it's super easy to understand. We'll be focusing on how to calculate the measures of angles AOB and BOC when they're supplementary and adjacent, and you're given their expressions in terms of 'x'. So, grab your calculators, and let's get started!
Understanding Supplementary and Adjacent Angles
Before we jump into the calculations, let's make sure we're all on the same page about what supplementary and adjacent angles actually mean. This is crucial for understanding the problem and solving it correctly. Trust me, getting the basics down is half the battle!
What are Supplementary Angles?
Supplementary angles are two angles that, when added together, equal 180 degrees. Think of it like this: they supplement each other to form a straight line. Imagine a straight line cut by another line – the two angles formed on one side of the line are supplementary. So, if you have two angles, let's say angle A and angle B, and if m∠A + m∠B = 180°, then angles A and B are supplementary. This is a fundamental concept, so keep it in mind!
What are Adjacent Angles?
Adjacent angles, on the other hand, are angles that share a common vertex and a common side, but they don't overlap. Think of them as angles that are right next to each other. Picture two slices of a pie – they share the pointy center (the vertex) and the line that separates them (the common side). So, if you have two angles that are side-by-side, sharing a vertex and a ray, they're adjacent. Got it?
Supplementary and Adjacent Angles Together
Now, when you combine these two concepts, you get supplementary adjacent angles. These are two angles that are both adjacent (sharing a vertex and a side) and supplementary (adding up to 180 degrees). This is exactly what we're dealing with in our problem, so it's super important to have a clear picture of what this looks like. Visualizing this will make solving the problem much easier. We're talking about two angles snuggled up next to each other, forming a straight line together. Cool, right?
Setting Up the Equation
Okay, now that we've nailed down the definitions, let's get to the nitty-gritty of our problem. We're given that angles AOB and BOC are supplementary and adjacent. We also know that m∠AOB = x - 11 degrees and m∠BOC = 2x + 11 degrees. The key here is to use the definition of supplementary angles to set up an equation. This is where the magic happens!
Using the Supplementary Angle Property
Remember, supplementary angles add up to 180 degrees. So, if angles AOB and BOC are supplementary, then their measures must add up to 180 degrees. We can write this as:
m∠AOB + m∠BOC = 180°
This is our foundation. This simple equation is the key to unlocking the solution. Once you've got this, you're on the right track.
Substituting the Given Expressions
Now, let's substitute the expressions we were given for the measures of angles AOB and BOC into our equation. We know that m∠AOB = x - 11 and m∠BOC = 2x + 11. So, we can replace those in our equation:
(x - 11) + (2x + 11) = 180
See how we've taken the information given and turned it into a concrete mathematical equation? This is a crucial step in problem-solving. We've translated the words into a language we can work with – math!
Simplifying the Equation
Before we solve for 'x', let's simplify the equation a bit. This will make the calculations easier and reduce the chances of making a mistake. Nobody wants to mess up on simple arithmetic, right?
Combine the 'x' terms: x + 2x = 3x
Combine the constant terms: -11 + 11 = 0
So, our equation simplifies to:
3x = 180
Boom! We've gone from a seemingly complicated problem to a simple algebraic equation. This is the power of breaking things down step by step. We're almost there, guys!
Solving for 'x'
Alright, we've got our simplified equation: 3x = 180. Now, let's solve for 'x'. This is a straightforward algebraic step, but it's essential to get it right. Remember, 'x' is the key to finding the measures of our angles.
Isolating 'x'
To isolate 'x', we need to get it by itself on one side of the equation. Right now, 'x' is being multiplied by 3. To undo multiplication, we need to divide. So, we'll divide both sides of the equation by 3:
(3x) / 3 = 180 / 3
Calculating the Value of 'x'
Now, let's do the division:
3x / 3 = x
180 / 3 = 60
So, we have:
x = 60
Yes! We found 'x'! This is a big step. But remember, we're not done yet. We need to use this value of 'x' to find the measures of angles AOB and BOC. Don't stop now, we're in the home stretch!
Calculating the Angle Measures
Okay, we've found that x = 60. That's awesome! But the problem asked us to find the measures of angles AOB and BOC, not just the value of 'x'. So, let's use our value of 'x' to calculate those angle measures. This is where everything comes together!
Finding m∠AOB
We know that m∠AOB = x - 11 degrees. Now that we know x = 60, we can substitute that value in:
m∠AOB = 60 - 11
m∠AOB = 49 degrees
There we go! We've found the measure of angle AOB. One angle down, one to go!
Finding m∠BOC
Next, we need to find the measure of angle BOC. We know that m∠BOC = 2x + 11 degrees. Again, we'll substitute x = 60:
m∠BOC = 2(60) + 11
m∠BOC = 120 + 11
m∠BOC = 131 degrees
Fantastic! We've found the measure of angle BOC. We've calculated both angles! High fives all around!
Verifying the Solution
Just to be extra sure we've got the right answer, let's quickly verify that our angles are indeed supplementary. Remember, supplementary angles add up to 180 degrees. So, let's add the measures of angle AOB and angle BOC:
49 degrees + 131 degrees = 180 degrees
Perfect! They add up to 180 degrees. This confirms that our solution is correct. Always a good idea to double-check your work, guys!
Conclusion
And there you have it! We've successfully calculated the measures of angles AOB and BOC, given that they are supplementary adjacent angles and their expressions in terms of 'x'. We broke down the problem step by step, from understanding the definitions of supplementary and adjacent angles to setting up the equation, solving for 'x', and finally, calculating the angle measures. We even verified our solution to make sure we were spot on.
Remember, the key to solving geometry problems (and really, any math problem) is to break it down into smaller, manageable steps. Understand the definitions, set up the equation carefully, and don't be afraid to double-check your work. You've got this!
I hope this explanation was helpful and cleared up any confusion you might have had. Keep practicing, and you'll be a geometry pro in no time! Now, go tackle some more awesome math problems!